Neighbor's tree - Security and dirt. Who is responsible?

You don't know how the newcomer was received by the old one. Maybe it also triggered a gag reflex

We were all not present at the conversation. Maybe the OP was also a bit upset because he saw his child in danger and therefore used a slightly inappropriate tone (from the neighbor's perspective). It is completely understandable that this is met with a defensive attitude, and most of us would probably do the same.

I don't like the trees either, but that was already known when the property was purchased. Therefore, no one can complain about it now.

As I said, I would leave everything as it is for now and continue with the garden planning. In the worst case, one of the trees or large branches might fall into the garden during the next storm. Unfortunate, but as long as the house is not hit, it is not a big deal. That is what insurance is for. The child will probably hardly be playing in the garden at that time.

We were all not present at the conversation.

Exactly. We were not there.
And since the OP doesn’t say anything or clarify anything to us, one might conclude that he probably recognized some mistakes in communication there, which are being pointed out to him here, and needs to let that sink in first. But we don’t know.

Probably applies to all pages: "A person, if he wants to ignore something,/ must not vergreifen in tone." (Eugen Roth). Most neighborhood conflicts are not about the matter itself.

It is expressed in a phenomenon that is rather modern to me, that the house builder sometimes has his fence up before the house is built. Of course, this is allowed, but for some it also expresses something. I think that here too things lie somewhat between white and black. Especially in the interpersonal area, "rights" are among the most difficult things to implement. Often (this does not have to be the case here) one also likes to see only one’s own right but not the duty, and that is usually seen by the other. Of course, there are also more or less unpleasant people.

Section 910 of the Building Code, paragraph 1 applies here. Set a reasonable deadline.

I would be cautious, paragraph 2 restricts this again, and it is not said whether the branches impair the usage.